Properties

Label 2-18e2-108.31-c2-0-1
Degree $2$
Conductor $324$
Sign $-0.826 + 0.562i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.376 + 1.96i)2-s + (−3.71 − 1.47i)4-s + (1.26 − 7.19i)5-s + (2.67 + 3.18i)7-s + (4.30 − 6.74i)8-s + (13.6 + 5.20i)10-s + (−18.4 + 3.24i)11-s + (−15.1 + 5.50i)13-s + (−7.27 + 4.05i)14-s + (11.6 + 10.9i)16-s + (−7.49 + 12.9i)17-s + (−0.338 + 0.195i)19-s + (−15.3 + 24.8i)20-s + (0.550 − 37.3i)22-s + (−10.6 + 12.7i)23-s + ⋯
L(s)  = 1  + (−0.188 + 0.982i)2-s + (−0.929 − 0.369i)4-s + (0.253 − 1.43i)5-s + (0.382 + 0.455i)7-s + (0.537 − 0.843i)8-s + (1.36 + 0.520i)10-s + (−1.67 + 0.295i)11-s + (−1.16 + 0.423i)13-s + (−0.519 + 0.289i)14-s + (0.726 + 0.686i)16-s + (−0.441 + 0.763i)17-s + (−0.0178 + 0.0102i)19-s + (−0.767 + 1.24i)20-s + (0.0250 − 1.69i)22-s + (−0.463 + 0.552i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.826 + 0.562i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00957919 - 0.0311274i\)
\(L(\frac12)\) \(\approx\) \(0.00957919 - 0.0311274i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.376 - 1.96i)T \)
3 \( 1 \)
good5 \( 1 + (-1.26 + 7.19i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (-2.67 - 3.18i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (18.4 - 3.24i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (15.1 - 5.50i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (7.49 - 12.9i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (0.338 - 0.195i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (10.6 - 12.7i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (36.9 + 13.4i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (14.0 - 16.6i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (-5.57 + 9.65i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-56.8 + 20.6i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-24.4 + 4.31i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-14.6 - 17.4i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 51.2T + 2.80e3T^{2} \)
59 \( 1 + (82.3 + 14.5i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-19.0 + 15.9i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (17.9 + 49.4i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (75.5 + 43.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-17.8 - 30.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.59 - 9.87i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (34.4 - 94.7i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (1.45 + 8.25i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39797307111748616607205249368, −10.82618862884036895201571703719, −9.674130549307255123196145679727, −9.058711173887934338112155046892, −8.064726296940252989938924170145, −7.45656115387463935248359649020, −5.81261135872696564166890926124, −5.17434889525323597794810172637, −4.36681204268424000193361718674, −1.95940203103513442641949639356, 0.01527186090047464571780493988, 2.36025646989401234928530669083, 2.99359832297582932505219372656, 4.53844789657806963154321855750, 5.69674082485548731875674325441, 7.38375122668825204741220212705, 7.79012593164571943659668745144, 9.322645963448818484297683132256, 10.27403067435368057632228790128, 10.74837231261040954556992427401

Graph of the $Z$-function along the critical line